Hi there, have been up the Sulphide a couple times this month and approach is pretty clear trail. Everything's melted out till you gain the ridge just inside park boundary. No need for the permit unless you're sleeping in there, and they usually go the day before on the weekends. We have camped just outside the boundary w no problems sans permit, though going C2C you'll have no issues.
We roped on the glacier, though these were both Mountaineers trips, so to be expected. The gully can be nasty w rockfall if parties are above you, and though I've wanted to, haven't done the ridge yet. Some place light protection. Downclimbing the gully is fine, though definitely some exposed moves. Half of our party chose to rappel last time.
Have fun!
Jason